Nodal solutions for the Kirchhoff-Schrödinger-Poisson system in $ \mathbb{R}^3 $
This paper is dedicated to studying the following Kirchhoff-Schrödinger-Poisson system: $ \begin{equation*} \left\{\begin{array}{ll} - \left(a+b \int_{ \mathbb{R}^3} |\nabla u|^2 dx \right) \Delta u+V(|x|) u+\lambda\phi u = K(|x|)f(u), & x \in \mathbb{R}^{3}, \\ -\Delta \phi = u^2,...
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| Format: | Article |
| Language: | English |
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AIMS Press
2022-07-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2022922?viewType=HTML |
| _version_ | 1811221850900922368 |
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| author | Kun Cheng Li Wang |
| author_facet | Kun Cheng Li Wang |
| author_sort | Kun Cheng |
| collection | DOAJ |
| description | This paper is dedicated to studying the following Kirchhoff-Schrödinger-Poisson system:
$ \begin{equation*} \left\{\begin{array}{ll} - \left(a+b \int_{ \mathbb{R}^3} |\nabla u|^2 dx \right) \Delta u+V(|x|) u+\lambda\phi u = K(|x|)f(u), & x \in \mathbb{R}^{3}, \\ -\Delta \phi = u^2, & x \in \mathbb{R}^{3}, \end{array}\right. \end{equation*} $
where $ V, K $ are radial and bounded away from below by positive numbers. Under some weaker assumptions on the nonlinearity $ f $, we develop a direct approach to establish the existence of infinitely many nodal solutions $ \{u_k^{b, \lambda}\} $ with a prescribed number of nodes $ k $, by using the Gersgorin disc's theorem, Miranda theorem and Brouwer degree theory. Moreover, we prove that the energy of $ \{u_k^{b, \lambda}\} $ is strictly increasing in $ k $, and give a convergence property of $ \{u_k^{b, \lambda}\} $ as $ b\rightarrow 0 $ and $ \lambda \rightarrow 0 $. |
| first_indexed | 2024-04-12T08:07:18Z |
| format | Article |
| id | doaj.art-f6cec41258b149fa8ec7a10297fa53cc |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-12T08:07:18Z |
| publishDate | 2022-07-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-f6cec41258b149fa8ec7a10297fa53cc2022-12-22T03:41:08ZengAIMS PressAIMS Mathematics2473-69882022-07-0179167871681010.3934/math.2022922Nodal solutions for the Kirchhoff-Schrödinger-Poisson system in $ \mathbb{R}^3 $Kun Cheng0Li Wang11. School of Information Engineering, Jingdezhen Ceramic University, Jingdezhen 333403, China2. College of Science, East China Jiaotong University, Nanchang 330013, ChinaThis paper is dedicated to studying the following Kirchhoff-Schrödinger-Poisson system: $ \begin{equation*} \left\{\begin{array}{ll} - \left(a+b \int_{ \mathbb{R}^3} |\nabla u|^2 dx \right) \Delta u+V(|x|) u+\lambda\phi u = K(|x|)f(u), & x \in \mathbb{R}^{3}, \\ -\Delta \phi = u^2, & x \in \mathbb{R}^{3}, \end{array}\right. \end{equation*} $ where $ V, K $ are radial and bounded away from below by positive numbers. Under some weaker assumptions on the nonlinearity $ f $, we develop a direct approach to establish the existence of infinitely many nodal solutions $ \{u_k^{b, \lambda}\} $ with a prescribed number of nodes $ k $, by using the Gersgorin disc's theorem, Miranda theorem and Brouwer degree theory. Moreover, we prove that the energy of $ \{u_k^{b, \lambda}\} $ is strictly increasing in $ k $, and give a convergence property of $ \{u_k^{b, \lambda}\} $ as $ b\rightarrow 0 $ and $ \lambda \rightarrow 0 $.https://www.aimspress.com/article/doi/10.3934/math.2022922?viewType=HTMLnodal solutionskirchhoff type problemschrödinger-poisson systemvariational methodsdisc's theorem |
| spellingShingle | Kun Cheng Li Wang Nodal solutions for the Kirchhoff-Schrödinger-Poisson system in $ \mathbb{R}^3 $ AIMS Mathematics nodal solutions kirchhoff type problem schrödinger-poisson system variational methods disc's theorem |
| title | Nodal solutions for the Kirchhoff-Schrödinger-Poisson system in $ \mathbb{R}^3 $ |
| title_full | Nodal solutions for the Kirchhoff-Schrödinger-Poisson system in $ \mathbb{R}^3 $ |
| title_fullStr | Nodal solutions for the Kirchhoff-Schrödinger-Poisson system in $ \mathbb{R}^3 $ |
| title_full_unstemmed | Nodal solutions for the Kirchhoff-Schrödinger-Poisson system in $ \mathbb{R}^3 $ |
| title_short | Nodal solutions for the Kirchhoff-Schrödinger-Poisson system in $ \mathbb{R}^3 $ |
| title_sort | nodal solutions for the kirchhoff schrodinger poisson system in mathbb r 3 |
| topic | nodal solutions kirchhoff type problem schrödinger-poisson system variational methods disc's theorem |
| url | https://www.aimspress.com/article/doi/10.3934/math.2022922?viewType=HTML |
| work_keys_str_mv | AT kuncheng nodalsolutionsforthekirchhoffschrodingerpoissonsysteminmathbbr3 AT liwang nodalsolutionsforthekirchhoffschrodingerpoissonsysteminmathbbr3 |